computation of the power flow solution of a radial distribution system for harmonic components

International Journal of Energy, Information and Communications

Vol. 4, Issue 1, February, 2013

37

Computation of the Power Flow Solution of a Radial Distribution

System for Harmonic Components

S. V. D. Anil Kumar1 and K. Ramesh Reddy

2

1Assoc. Professor, Department of EEE, St. Ann’s College of Engg. and Tech., Chirala,

Andhra Pradesh, India 2HOD & Dean of PG Studies, GNITS, Hyderabad, A.P., India

[email protected], [email protected]

Abstract

The paper proposes a novel algorithm to compute the power flow solution of a Radial

Distribution System (RDS) accounting for all the harmonic components. It uses a recursive

solution technique. The proposed method uses a novel Dynamic Data Structure reported in

the paper. The proposed method is tested upon a 33-bus RDS and the results are reported.

Radial distribution systems (RDS) require special load flow methods to solve power flow

equations owing to their high R/X ratio. Increasing use of power electronic devices and effect

of magnetic saturation cause harmonics in RDS.

Keywords: Radial Distribution Systems, Magnetic saturation, Harmonics, Dynamic Data

Structure Nomenclatures

Pij(k) kth harmonic component of real power flowing in the line at the sending end

PLij(k) Real power losses in the transmission line for the kth order of harmonics

Qij(k) kth harmonic component of reactive power flowing in the line at the sending

end

QLij(k) Reactive power losses in the transmission line for the kth order of harmonics

rij Resistance offered to the kth harmonic power flow between buses i and j

Vi(k) Sending end bus voltage for kth order of harmonics

Vj(k) Receiving end bus voltage for kth order of harmonics

xij(k) Reactance offered to the kth harmonic power flow between buses i and j

Greek Symbols

i(k) Angle of sending end bus voltage for kth order of harmonics

j(k) Angle of receiving end bus voltage for kth order of harmonics

(k) Acceptable tolerance value for the power mismatch of kth order of harmonics

1. Introduction

Analysis of distribution systems using power flow is important in the field of power

systems. Distribution systems are predominantly characterized by their high R/X ratio and

radial topology. Matrix based iterative methods do not lend themselves for radial distribution

systems owing to these characteristics. Numerous algorithms have been developed using

simple recursive equations [1-3]. Rapid industrialization has lead to increasing use of power

electronic devices in transmission and distribution systems. Modern industrial and domestic



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consumers use an ever-increasing number of devices that primarily employ power electronics

based power-conditioners. Use of AC machines employing magnetic circuits in the saturated

region also introduces harmonics in electric power systems. Numerous power flow methods

have been reported in literature that is meant to handle harmonics [4]. These methods seldom

address radial distribution systems.

Load flow calculation in harmonic polluted radial system with distributed generation has

been carried out using abstract data types with complex parameters [5]. A multiple-frequency

three-phase load-flow with two sub models including the fundamental power flow (FPF) and

harmonic frequency power-flow (HPF) model has been developed and the standard Fourier

analysis was used to deal with the harmonic loads to get injection currents [6]. Fuzzy

number based methodology for harmonic load-flow calculation including uncertainties has

been applied for interconnected system [7]. From the above, one may see the need for an

efficient algorithm that reliably solves the power flow equations for radial distribution

systems characterized by high R/X ratio, radial topology and harmonic loads. Large

distribution systems employ Supervisory Control and Data Acquisition (SCADA) systems for

efficient management. SCADA systems employ power flow solutions methods to ascertain

the state of the distribution system. Distribution Load Flow (DLF) also forms an integral part

of algorithms that assess the cost and benefit of transformer taps changes, change in static var

settings, reconfiguration of the system for various purposes ranging from load balancing,

loadability enhancement, distribution loss minimization, and voltage profile improvement

amongst several others. Existing methods of distribution load flow utilize look up tables

and/or load tables and/or switch tables for the purpose of system representation.

Reconfiguration and other actions render the look up table based representation schemes

ineffective.

Artificial Neural Network (ANN) approach has been applied for harmonic load flow

analysis of a distribution system [8].

This paper proposes a Dynamic Data Structure (DDS) that helps to store information of a

branch of a RDS to determine the bus voltages and angle of a particular load pattern for a

given order of harmonics. This DDS is adaptation of the DDS reported in [9]. These DDS are

handled as a linked list representing the entire radial distribution system. A pseudo code that

generates the DDS for a RDS is presented. A function is then developed that computes the

voltages from the farthermost end up to the head of the branch. This function is called

recursively to find out voltages at all branches of the RDS from the farthermost branch up to

the branches emanating from the main substation of the RDS. A pseudo code for this

recursive function is also presented. The resulting DLF algorithm is computationally efficient

and can deal with the topology changes quickly. The proposed method allows modelling of

loads of any type, any number of harmonic components and radial system of any

configuration with respect to buses and branches.

2. RDS Representation Using Dynamic Data Structure

Consider a radial distribution system shown in Figure 1. This system represents typical

RDS. It has several branches and buses. The dynamic data structure must represent the RDS

or an integral part of the RDS. The proposed DDS represents an integral part of the RDS,

namely a generic branch. This section presents the details of the proposed novel DDS that is

used to store details of a branch.

The following attributes were envisioned while developing the proposed DDS.

(a) The data structure must be dynamic in nature so that it may be created and altered in the

execution stage.



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(b) DDS required for holding the information of a branch must be compact and addressable

from any function.

(c) It must be flexible to accommodate any number of buses and harmonic loads within a

branch.

(d) It must be flexible to address as many data structures of branches that emanate from the

end of a branch.

Figure 1. A Simple Radial Network with Several Branches

The proposed data structure of a branch needs to hold the details of:

(a) Parent bus to which the branch is connected,

(b) Number of buses in the branch.

(c) An array to store IDs of all the buses in the order of their location from the head of

the branch,

(d) Details of line resistance and reactance for various harmonic components with in a branch,

(e) Number of branches emanating from the last bus of this branch,

(f) A list of pointers pointing to the data structures that store information of

emanating branches, and

(g) An array to store IDs of buses at the head of emanating branches.

The definition of the data structure is given below.

Typedef struct {

Integer parent-bus-id;

Integer number-of-buses-in-branch, array-of-bus-ids;

Float resistance and reactance for all harmonic components of lines from

each bus to a bus towards head bus;

Integer number-of-branches, array-of-head-bus-id-of- emanating-branches;

void pointers-to-structure-of-emanating-branches;

} branch;

The pictorial representation of the proposed DDS is shown in Figure 2. Starting from the

main substation, branches are sequentially stored in the data structure using a recursive

function that is outlined in the following pseudo code:



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void create_ structure(first bus of the branch, address

(pointer to) of the data structure, parent bus ID number from

which this branch emanates)

{

temp = temporary array to store the IDs of buses in this

branch;

number-of-buses-in-branch = 1

temp[first location] = bus-ID = head bus ID

while ( number of branches emanating from bus-ID = 2)

{

increase number-of-buses count by 1;

store this bus-ID in temp array;

bus Id changes = next connected bus ID;

}

Dynamically define array-of-bus-IDs to hold bus IDs of buses in

the branch;

Store the content of temp in array-of-bus-IDs;

Dynamically define array-of-r-x to hold resistance and reactance

for various harmonics of lines between each bus and another

leading to the head bus in the branch;

Fill these arrays with appropriate values based upon the loads

encountered;

Determine number-of-branches emanating from the last bus this

branch -> nbr;

If nbr > 0

{

dynamically define array-of-head-bus-id-of-emanating-branches

and fillup;

dynamically define pointers-to-structure-of-emanating-

branches;

allocate space for nbr data structure and store their addresses

in above;

for each of the nbr emanating branches -> call function

create_structure(); } }

Figure 2. Proposed RDS with nh Harmonics



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The pseudo code presented above creates a structure for a branch. Starting at the main

substation bus, the pseudo code forms the first branch by sequentially considering buses

until it finds one that has several branches emanating from it.

Referring to Figure 1 that presents a sample radial system, it builds the first branch from bus

1 up to bus 3 where two branches emanate. The data structure created for branch labelled I,

starts with bus 01 and ends with bus 03 comprising three buses. It has two branches

emanating from it namely II and III. This structure stores pointers that point towards the

structures storing details of branches II and III. The pseudo code calls itself twice, once each

for branches II and III. This process proceeds until data structure for all the branches are built.

A pictorial representation of the data structure storing details of branch II is shown in Figure 3.

The DDS proposed in this paper is highly flexible and is convenient to use when the

RDS is reconfigured under the umbrella of SCADA. This build up of DDS to represent the

entire RDS is convenient for the solving the power flow equations for the entire radial

network using a recursive function outlined in the next section.

Figure 3. Example of Dynamic Data Structure – Branch II

3. Proposed Harmonic Distribution Load Flow

Distribution power flow is presented in this section. It uses the dynamic data structure

proposed in Section 2. First the line model of a generic line is presented with modelling for

harmonics. Then a recursive algorithm is presented. This is followed by the flowchart of the

proposed harmonic distribution load flow. 3.1. Line model and voltage equation considering harmonics

In this section, a simple circuit model of a transmission line considering kth order

harmonic component and associated recursive voltage equations is presented. It is assumed

that the three-phase RDS is balanced and can be represented by an equivalent single-phase

system. The transmission line to ground capacitance elements at the distribution voltage level

are small and thus neglected. A simple circuit model of a transmission line is shown in Figure

4.

Figure 4. Simple Equivalent Circuit of Transmission Line Considering Harmonics



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The values of Vi(k)i(k) and Vj(k)j(k) represent the sending and receiving end voltages. The

values rij + jxij(k) represent resistance and reactance offered to the kth harmonic power

flow between buses i and j. Several methods of solving the power flow equations of

distribution systems have been proposed [1-3]. It may be observed that these methods use

recursive equations as below in several forms considering either sending or receiving end

power (Figure 4 for the variables used):

(a) Equation derived considering sending end powers of kth harmonic:

where Pij(k) and Qij(k) refer to the kth harmonic component power flowing in the line at the

sending end.

(b) Equation derived considering receiving end powers of kth harmonic:

where Pij(k) and Qij(k) refer to the kth harmonic component of power flowing in the line at the

receiving end. These equations are not amenable for matrix computation as in the case of

conventional methods of solving power flow equations for transmission grids like NR method

or Fast De-coupled Load Flow method.

3.2. Line Model and Voltage Equation Considering Harmonics

The proposed recursive algorithm to compute the voltage solution of the RDS is presented

in this section. The routine starts by computing voltage from the farthermost bus of the first

branch. If the branch has other branches emanating, the pseudo code recursively calls itself to

compute the state of buses in these branches and the power they draw from the farthermost

bus. This recursive call is continued until the branches from where other branches do not

emanate. Then, the pseudo code computes the voltage at farthermost bus using (2) with the

knowledge of its load. Then using the expressions below, it computes the power loss in the

transmission line connecting this bus and next bus in the direction leading to the first bus

of the branch.

where PLij(k) and QLij(k) are the real and reactive power losses in the transmission

line model shown in Figure 4 and Pij(k) and Qij(k) refer to the power flowing in the

line at the receiving end corresponding to the kth harmonic component. With load at

bus j and transmission power loss in the line between buses i and j known, the pseudo

code computes the load at bus i. This process continues until the voltage is computed until

the first bus. The phase angle of the voltage phasor at the jth bus is computed by the

following expression:



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where Pij(k) and Qij(k) refer to the power flowing in the line at the receiving end

corresponding to the kth component. During each computation, it repeats the solution for

each harmonic component.

In this work convergence is checked by ascertaining whether the sum of powers flowing

out in the lines connected to each bus equals, or nearly equals within a tolerable limit, the net

power injected in to that bus by the connected generations and loads. Mathematically,

convergence criterion is represented as

Where E(k) is an acceptable tolerance value kth harmonic.

3.3. Overall Algorithm

This section presents the details of the proposed method that uses the novel data structure

and the proposed recursive algorithm. The overall algorithm is presented in Figure 5. The

algorithm reads in the data at first. It then creates one DDS each to represent all the branches

of the RDS. The downstream harmonics are considered to generate appropriate reactance

values of all the lines. The DDS and pseudo code required are presented in Section 2.

Subsequently, the algorithm calls the function calculate load/generation () recursively to

compute the voltage solution at all the buses in the system. The algorithm checks for

convergence by checking whether the inequalities (5) and (6) are satisfied. The function

calculate load/generation () is repeatedly called until the inequalities (5) and (6) are satisfied.

In the implementation of the overall algorithm, transformers are modelled using conventional

π-model. The effect of shunt compensating element at any bus is considered. π-model of a

transformer is used for all computations.

Figure 5. Flowchart for Overall Algorithm



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4. Results

To test the effectiveness of the proposed algorithm 33-bus is considered. The single line

diagram of the 33-bus RDS is presented in Figure A-1 (Appendix A) Three harmonic

components namely 3, 5 and 7 are considered for the purpose of simulation. The tolerance

was chosen to be 0.001p.u. It took seven iterations for the proposed algorithm to converge in

for the 33-bus RDS. The convergence depends upon the base load condition and the chosen

tolerance. The proposed method is expected to give the power flow solution even for large

systems with minimum number of iterations. Table A-1 (Appendix A) presents the results of

the proposed method with voltage solution for various harmonics. Table A-2 presents

comparison of base case solution with solution from ladder iterative technique and from BPN

[8]. Figure A-2 (Appendix A) presents a graph that shows total system loss variation with

respect to harmonic components.

5. Conclusions

Inclusion of power electronic devices and saturation of magnetic circuits introduce several

load side harmonics. This paper reports a new distribution system load flow algorithm that

uses recursive voltage equations considering harmonic load components. This paper proposes

a new dynamic data structure for modular representation of a radial distribution system with

loads having harmonic component. The DDS are stored and retrieved using a linked list. A

recursive load flow algorithm is then proposed that uses the dynamic data structure to solve

the power flow equations efficiently considering harmonic components. Pseudo codes for

generation of dynamic data structure are included. Results of the tests on a 33-bus RDS with

harmonic components are presented that demonstrates the applicability of the method.

References [1] W. H. Kersting and D. L. Mendive, “An application of ladder network theory to the solution of three phase

radial load flow problems”, Proceeding of IEEE PES, Winter Power Meeting Conference, New York, USA,

(1976).

[2] W. H. Kersting, “A method to design and operation of distribution system”, IEEE Transactions on

Power Apparatus and systems, vol. PAS-103, no. 7, (1984), pp. 1945-1952.

[3] M. E. Baran and F. F. Wu, “Optimal sizing of capacitors placed on a radial distribution system”, IEEE

Transactions on Power Delivery, vol. 4, no. 1, (1989), pp. 735-743.

[4] S. Herraiz, L. Sainz and J. Clua, “Review of harmonic load flow formulations”, IEEE Transactions on Power

Delivery, vol. 18, no. 3, (2003), pp. 1079–1087.

[5] C. Bud, B. Tomoiaga, M. Chindris and A. S. Anderu, “The load flow calculation in harmonic polluted

radial electric networks with distributed generation”, Proceedings of the 9th International conference on

Electrical power quality and Utilisation, Barcelona, (2007).

[6] W. M. Lin, T. S. Zhan and M. T. Tsay, “Multiple-frequency three phase load flow for harmonic analysis”,

IEEE transactions on Power Systems, vol. 19, no. 2, (2004), pp. 897-904.

[7] A. A. Romero, H. C. Zini, G. Ratta and R. Dib, “A fuzzy number based methodology for harmonic load-flow

calculation, considering uncertainties”, Latin American Research, vol. 38, no. 3, (2008), pp. 205-212.

[8] A. Arunagiri, B. Venkatesh and K. Ramasamy, “Artificial neural network approach-an application to radial

load flow algorithm”, IEICE Electronics Express, vol. 3, no. 14, (2006), pp. 353-360.

[9] B. Venkatesh and R. Ranjan, “Data structure for radial distribution system load flow analysis”, IEE

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Appendix A

Figure A-1. 33-BUS Radial Distribution System

Figure A-2. Graph of variations in losses with Harmonic



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Table A-1. Results of 33-Bus System with Fundamental and Three Harmonic Components



47



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Table A-2. Comparison of Load Flow Results for the Base Case with the Proposed Method



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Authors

Dr. K. RAMESH REDDY received B.Tech. in Electrical and

Electronics Engineering from Nagarjuna University, Nagarjuna

Nagar, India in 1985, M.Tech in Electrical engineering from National

Institute of Technology (Formerly Regional Engineering College),

Warangal, India in 1989, and Ph.D from SV University, Tirupathi,

India in 2004. Presently he is Head of the department and Dean of

PG studies in the Department of Electrical & Electronics Engineering,

G.Narayanamma Institute of Technology & Science (For Women),

Hyderabad, India. Prof. Ramesh Reddy is an author of 16 journal and

conference papers, and author of two text books. His research and

study interests include power quality, Harmonics in power systems

and multi-Phase Systems.

S. V. D. ANIL KUMAR received B.Tech. in Electrical and

Electronics Engineering from SV University, Tirupathi, India in 2000,

M.Tech in Power Electronics from Visveswaraiah Technological

university, Belgaum,India in 2005. He is Ph.D Student at Department

of Electrical Engineering, JNT University, Hyderabad, India.

Mr.S.V.D. Anil Kumar is an author of 5 journal and conference

papers. His research and study interests include power quality,

Harmonics in Power Systems.



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computation of the power flow solution of a radial distribution system for harmonic components

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